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PHYSICAL REVIEW LETTERS
PRL 102, 108301 (2009)
Particle Pressure in a Sheared Suspension:
A Bridge from Osmosis to Granular Dilatancy
Angélique Deboeuf, Georges Gauthier, and Jérôme Martin
Univ Pierre et Marie Curie-Paris6, Univ Paris-Sud, CNRS, Laboratoire FAST, Campus universitaire d’Orsay,
91405 Orsay Cedex, France
Yevgeny Yurkovetsky and Jeffrey F. Morris
Levich Institute and Chemical Engineering, City College of New York, New York, New York 10031, USA
(Received 10 July 2008; published 9 March 2009)
The normal stress exerted by particles in a sheared suspension is measured by analogy with a method
used to measure osmotic pressure in solutions. Particles in a liquid are confined by a fine screen to a gap
between two vertical concentric cylinders, the inner of which rotates. Pressure in the liquid is sensed either
by a manometer or by a pressure transducer across the screen. The particles are large enough so that
Brownian motion and equilibrium osmotic pressure are vanishingly small. The measured pressure yields
the shear-induced particle pressure , the nonequilibrium continuation of equilibrium osmotic pressure.
For volume fractions 0:3 0:5, is strongly dependent on , and linear in shear rate. Comparisons
of the measured particle pressure with modeling and simulation show good agreement.
DOI: 10.1103/PhysRevLett.102.108301
PACS numbers: 83.80.Hj, 83.80.Fg
Motion of one component relative to the remainder
of a mixture may arise from different sources, including
a density difference or concentration gradient. Driving
force is often described in terms of a potential, e.g.,
the gravitational potential energy for settling particles.
However, diffusion is usually accounted through a
Fickian flux, j ¼ Drn, with D the diffusivity and n
the solute number density. Because diffusion results from
irregular stochastic motion, defining an associated potential is not trivial. The desired potential is, however, obtained from the relation between diffusion and osmotic
pressure and this is considered as a prelude to a striking
extension of osmotic pressure in flowing mixtures.
Random motion of dissolved molecules and ions, or of
submicron particles, results in diffusion scaling as the thermal energy kT. In the absence of other influences, diffusion
tends to drive the mixture toward the homogeneous equilibrium state with rn ¼ 0. A key development was provided by Einstein [1], who considered a dilute dispersion of
particles in an external field of potential energy U. The
force F ¼ rU, applied to each particle, induces a flux
nu with the velocity given by u ¼ MF where M is the
mobility of the solute particles. In the resulting inhomogeneous—but nonetheless equilibrium—state, the flux due to
the external field balances the diffusive flux so that
nMrU Drn ¼ 0. Requiring Boltzmann distribution of energy at equilibrium, n exp½U=kT, yields
both the thermodynamic force Fth ¼ kTr lnn and the
celebrated result D ¼ kTM. Moreover, as pointed out by
Einstein, the volumetric thermodynamic force nFth can be
written as rosm , where osm is the osmotic pressure
which obeys the van’t Hoff relation for dilute systems [2],
osm ¼ nkT. Hence, the Fickian flux may also be written
j ¼ Mrosm . Diffusive flux and osmotic stress are then
0031-9007=09=102(10)=108301(4)
two complementary descriptions to account for the influence of random motion of a solute. These motions cause
diffusive mixing but also tend to minimize variations of the
Helmholtz free energy A, leading to a thermodynamic
@A
definition of the osmotic pressure, osm ¼ ð@V
ÞN;T ,
with N the number of solute particles.
This thermodynamic approach provides a convenient
means to describe osmosis, which occurs when a solution
is separated from a lower concentration solution or the pure
solvent by a ‘‘semipermeable membrane’’ allowing the
flow of the solvent but restricting the solute to the solution
side, as in a U tube osmometer sketched in the left of Fig. 1.
In this context, osm is the additional pressure which
must be applied to the solution side to stop solvent flow,
i.e., to reach the equilibrium state. Note that this solvent
flux was first observed in 1748 [3], but its interpretation as
a decrease of the free energy by dilution was proposed
much later. Osmotic driving forces can be large: Sea water
contains ions at n 1:4 mol=L (2.5 mol %) and has
Thermal agitation
Shear
Shear rate
ω
∆h
∆h
solvent
suspending fluid
solution
∆h
wet
granular
media
suspension
grid
membrane
FIG. 1 (color online). Left: U tube osmometer. Center:
Schematic of the experimental apparatus. Right: Reynolds dilatancy.
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Ó 2009 The American Physical Society
PRL 102, 108301 (2009)
PHYSICAL REVIEW LETTERS
osm 25–30 atmospheres, surprisingly large yet in line
with the van’t Hoff relation for dilute solutions.
Here we illustrate that unexpected osmotic phenomena
occur in sheared suspensions of large particles. We recall
first that for hard spheres, osm ¼ nkT½1 þ 4gð2Þ
where the volume fraction is ¼ 4na3 =3 for spheres
of radius a, and gð2Þ is the contact value of the pair
distribution function [4]. Away from the divergence of g
at maximum packing, i.e., for concentrations smaller than
max 0:64, this expression yields osm kT=a3 , of
Oð1Þ Pa for a ¼ 0:1 m and Oð103 Þ Pa for a ¼ 1 m.
For larger particles, osmotic pressure and Brownian motion are then vanishingly small, but these ‘‘noncolloidal’’
suspensions nevertheless exhibit diffusion in sedimentation or shear [5,6], induced by the chaotic particle trajectories [7] caused by hydrodynamic interactions.
The interactions generate ‘‘shear-induced diffusion’’
which dissipates concentration gradients [8,9]. Rather remarkably, particle interactions driven by a spatially vary_ can generate concentration gradients. For
ing shear rate, ,
unidirectional flow ux ðyÞ this implies a cross-stream par_ where _ ¼ jdux =dyj [5]. As an alterticle flux j r,
native, a ‘‘particle pressure’’ has been invoked as the
driving potential for particle migration [10,11]. Particle
pressure and shear-induced diffusion are the counterparts
in sheared systems of the thermal osmotic pressure and
Brownian diffusion, respectively.
Our purpose is to further develop the connection, described in theory and simulation studies [12,13], between
nonequilibrium particle pressure and osmotic pressure.
In particular, we illustrate that may be measured using a
method directly analogous to the classical approach for the
measurement of osmotic pressure. The experiments show
that osmosis in mixtures is a general result of constraining
the dispersed phase against spreading. This leads to the
notion that quantifies the general tendency of a dispersed phase to spread, an idea long associated with
Reynolds dilatancy [14] in which a granular assembly,
with grains in contact, must expand to shear.
We turn now to the experimental method. When solute
or particles are constrained from spreading, stresses arise
in the mixture leading to solvent flow into the solution.
Osmotic pressure, associated with the drive toward solute
spreading and hence a compressive stress in the dispersed
component, is thus measured as a reduced pressure in the
solvent. We use this as the conceptual basis to measure the
pressures induced by shear flow in the two phases of a
viscous suspension. Shear-induced particle pressure has
eluded direct measurement in large part because opposing
stress states of solvent and dispersed phase lead to a
cancellation [13,15]. To measure requires, as for
osm , a method which either allows relative motion so
the dispersed phase may change its concentration, or alternatively directly senses the driving force for relative motion. We describe both approaches.
We use suspensions of polystyrene spheres (Dynoseeds
TS; Microbeads) of diameter 40, 80, or 140 m (5%,
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with AFM-measured roughness 100 nm), with density
¼ 1:05 g cm3 matching the liquid. The viscosity of the
Newtonian fluid, poly(ethylene glycol-ran-polypropylene
glycol) monobutyl ether, is f ¼ 3 Pa s at 20 C. The bulk
suspension viscosity agrees well with the form ðÞ=f ¼
ð1 =max Þ2 . The ratio of shear to Brownian mo_ 3 =kT ¼ Oð108 Þ
tion, or Péclet number, is Pe ¼ 6f a
1
at _ ¼ 1 s . Thermal motion is negligible.
The analogy of the device used here to a U tube osmometer with semipermeable membrane is illustrated in the
center sketch of Fig. 1. The actual Couette device is shown
at right in Fig. 2. The inner and outer cylinders have radii
r1 ¼ 17:5 mm and r2 ¼ 20 mm, respectively, leaving an
annular gap r2 r1 ¼ e ¼ 2:5 mm. Rotation of the inner
cylinder generates a shear rate _ r1 =e. Variation of
shear rate across the gap is small and we report only the
average. The flow is viscous as the Reynolds number is
_ 2 =f 1. Eight holes of diameter 3 mm pass
Re ¼ e
through the outer stationary cylinder. They are at 90
intervals, in two sets located 2 and 6 cm from the bottom
of the sheared annulus. Screens with 20 m square openings, small enough to retain the particles, are placed across
the holes on the side toward the annulus. A tube is fixed in
each hole and filled with the suspending liquid. These
tubes may serve as manometers or for transmission of the
liquid pressure to a transducer.
Results from the manometer approach for ¼ 0:4 and
_ ¼ 70 s1 are illustrated by Fig. 2. Meniscus levels are
shown for a screened hole before shearing (left tube) and
after 40 min of shearing, in the top and bottom images,
respectively. To demonstrate the role of the screen, the
meniscus in a tube attached to an unscreened hole is also
FIG. 2 (color online). Manometer approach for a suspension of
80 m diameter particles at ¼ 0:4. Couette device is at right,
with the upper surface of the suspension indicated by the dashed
line. Meniscus levels are highlighted by arrows for screened (left
tube) and open (right) holes at rest (top picture), and after 40 min
of shear at _ ¼ ri =e ¼ 70 s1 (bottom picture).
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PHYSICAL REVIEW LETTERS
shown in Fig. 2 (right tube). Whereas the meniscus associated with the screened hole drops a few cm, the one with
the unscreened hole is almost stationary, as liquid suction
is balanced by solid phase dispersive pressure.
Although visually convincing and appealing for its analogy with the U tube experiment, the manometer approach
has drawbacks. The long times (hours) needed to obtain a
measurement, and large h when and _ are both large
render the approach inconvenient. A basic flaw is that the
results are sensitive to , which is altered by the liquid
drawn into the annulus. The manometer measurements are
consistent with the approach described below.
To obtain quantitative measurements, a pressure transducer is attached to each tube. The screen is maintained,
and the tubing is filled with suspending liquid to provide
communication between suspension and transducer. A preshear was applied to ensure the suspension was uniform.
The transducer reading stabilizes within seconds of shearing, and measurements are averaged over 1–2 min. No
liquid crosses the screen. The difference in the transducer
reading between sheared and static states is P, and is
negative: Flow generates a suction pressure in the liquid.
The pressure is averaged over forward and reverse shearing
to eliminate flow-induced pressure due to any slight eccentricity of the cylinders.
Results for the suspension of 40 m diameter particles
_
are shown in the top panel of Fig. 3 as a function of f ,
with the fluid viscosity f corrected for temperature. The
measured jPj is viscously driven, as inertial effects
would be greatest for the pure liquid which gives a null
response, P ¼ 0, and jPj increases roughly linearly
_ Note that linearity in _ at Pe 1 differs
with increasing .
from the _ 2 dependence of normal stress observed for
small Pe [16]. The magnitude increases strongly with ,
from jPj=f _ ¼ 0:35 at ¼ 0:3 to 18.5 at ¼ 0:5.
Error bars are significant, but are typically small compared
to the mean value. Data for the other sizes of particles
studied (80 and 140 m) are similar to the 40 m case,
although P is smaller for the larger particles.
_ for
The measured suction pressures, normalized by f ,
all and particle sizes studied are gathered in the lower
panel of Fig. 3. These show that the suction pressure
measured in the liquid behaves similarly to the particle
normal stress differences [11,17]. In particular, the pressure becomes reliably measured at 0:3 and grows
rapidly with . A slight radial particle migration, which
can be deduced from results in [11], is complete during the
preshear for all a, and cannot explain the difference in jPj
with particle size. A particle slip layer [18] will not explain
the observation, although there may be a difference in
organization when the gap is small (e=2a < 20 for the
140 m beads) which could enhance apparent slip (as in
shear banding). We note that an axial migration to, and
even through, the free surface was observed and this reduces within the suspension; migration is more pronounced for larger beads. In general, these considerations
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FIG. 3 (color online). Measured difference in liquid pressure
between resting and sheared states, P. Top: 40 m diameter
_ Error bars
particle suspension at varying , as a function of f .
show the range of transducer measurements for a given condition. Bottom: jPj=
_ f as a function of for three particle
sizes.
lead us to expect that small particles yield the most accurate results, in accord with agreement with model and
simulation for the 40 m suspension (see Fig. 4).
There is a conceptual difficulty in the measurement of
the particle pressure in a sheared suspension, stemming
from the fact that liquid and solid both satisfy incompressibility to a close approximation. Particle pressure is thus
offset by an equal and opposite change in liquid pressure
_ A transducer exposed to both
[13], i.e., ¼ PðÞ.
phases measures the sum of these shear-induced stresses
to yield a confusing result [15]. Hence, only a method
which discriminates between the phases can enable the
measurement of the particle pressure, and in our approach
we make use of ¼ P ¼ jPj. This use of the liquid
pressure provides the conceptual link to osmotic pressure.
The particle pressure [12] in a sheared suspension is
FIG. 4 (color online). Comparison of experimental correlation,
simulation, and modeling of the particle pressure with the
experimental values obtained in this study taking P .
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PRL 102, 108301 (2009)
PHYSICAL REVIEW LETTERS
the mean normal stress exerted by the particle phase
¼ ð1=3Þ½P11 þ P22 þ P33 and has been shown by
simulation to be equivalent to the osmotic pressure for
¼ osm .
vanishing shear rate [13], i.e., lim!0
_
One has to be cautious in equating the measured liquid
pressure with the particle pressure, i.e., ¼ jPj. The
measured liquid pressure is set by the conditions at the free
surface of the suspension, where deformation of the liquid
surface allows capillary pressure to balance the mean particle stress in the vertical direction. It follows jPj ¼ P33 ,
which may differ from . Several arguments justify
the approximation jPj. First, simulation [13] and
experimental correlation [17] lead to the conclusion
that normal stress differences, N1 ¼ P11 P22 and
N2 ¼ P22 P33 , are small relative to . Furthermore,
the small change in meniscus level for an unscreened
hole (Fig. 2) confirms that Prr þ P ¼ P22 P33 0.
The particle pressure measured, using the approximation ¼ P, is compared in Fig. 4 with experimental
correlation, simulation, and modeling. Analysis of experimental data [17] showed that migration and viscous resuspension could be rationalized by an isotropic particle
stress. The resulting correlation developed for is plotted.
_ from constitutive modThe expression n ðÞ ¼ =f ,
eling for migration analysis [11] is also displayed, together
with simulation values of at Pe ¼ 1000 [13] and infinite
Pe [19], obtained with weak and zero thermal motion,
respectively. Both simulations use Stokesian Dynamics
but differ in the treatment of . Despite the variation
observed for the different particle sizes, our measured
particle pressure is in remarkable agreement with simulation and experimental correlation.
We conclude that liquid suction pressure in a sheared
suspension provides a robust means of measuring the particle pressure. The fundamental link between particle pressure and equilibrium osmotic pressure is strong. Bagnold’s
notion of particle pressure under shear [20] has a central
role in the modeling of shear-induced migration.
The utility of f _ for modeling is recognized and
applied for noncolloidal [12,13] and Brownian dispersions
[21,22]. Simulations have shown that the quantity agrees
with equilibrium theory [13]. The present work shows
direct measurability of , which, these models have argued, drives shear-induced migration flux [10–13]:
@
@
j r P r ¼ r þ
r_ : (1)
@
@_
Thus, particle stress can be seen to play a similar role both
in and out of equilibrium. The shear-induced migration
driven by r_ generates gradients in , but nonetheless
relaxes variations in , just as Fickian diffusion relaxes
variations in osm arising from nonzero r. One can then
speculate that particle pressure pertains to a more general
property of mobile particle mixtures, which exhibit rather
nonintuitive phenomena when the dispersed phase is confined. The phenomena include the well-known osmosis,
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observed in thermally-driven systems, and known to be
related to the osmotic pressure.
Particle pressure is the nonequilibrium continuation of
osmotic pressure [13] and drives shear-induced migration
in pipe and channel flows [10,23]. Because density matching is possible, for viscous suspensions can be explored
for any , and for arbitrary ratios of shearing to thermal
motion. When gravity plays a prominent role, solids loading is not freely variable. Hence, a granular medium or
sediment is formed with max and particles in contact. Under such conditions, shearing will induce dilation
and the granular material will develop normal forces if it is
sheared while constrained to the same volume. While the
final link with granular dilation remains to be made, particle pressure appears to be a bridge from osmotic pressure
driven by kT to the athermal mixture stresses in suspensions and granular materials.
We thank RTRA Triangle de la physique for funding,
Univ. de Paris 6 for a visiting professorship (J. F. M.),
A. Aubertin, and R. Pidoux for building apparatus, and
D. Bonamy and G. Pallares for the AFM measurements.
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